nLab rough paths

Contents

Idea

Rough Paths are a way of defining Riemann-Stieltjes type integrals for paths with poor pp-variation where Ito techniques don’t work in the context of solving controlled differential equations.

Set-Up

We want to solve differential equations of the form:

y t=y 0+ 0 tV(y s)dX s y_t=y_0+\int_0^t V(y_s)\otimes dX_s

where V=(V 1,...,V d)V=(V_1,...,V_d) with V iC b ( d, d)V_i\in C_b^\infty(\mathbb{R}^d,\mathbb{R}^d) smooth bounded vector fields and XC p([0,T], d)={f:f pvar:=(sup 𝒫|f tf s| p) 1/p<}X\in C^p([0,T],\mathbb{R}^d)=\{f\colon \|f\|_{p-var}\colon = (\sup_{\mathcal{P}} |f_t-f_s|^p)^{1/p}\lt \infty\} is a signal with finite pp variation.

We want to solve this by a fixed point argument/Picard iteration. That means we must define:

0 tf(X s)dX s \int_0^t f(X_s)\otimes dX_s

for smooth bounded ff.

The classical paper (Young 1936) says, in short, that one can define this integral as a Riemann-Stieltjes integral iff p<2p\lt 2.

A theorem by Bichdeller-Dellecherie says, in short, that one can define this integral in Itô sense? if and only if XX is a semimartingale?.

Main Result

Given a sequence of partitions, 𝒫 n={0=t 0<...<t n1<t}\mathcal{P}_n=\{0=t_0\lt...\lt t_{n-1}\lt t\}, if one can define X su 1:= s udX r 1X^1_{su}\colon = \int_s^u dX_{r_1}, X su 2:= s u s r 2dX r 1dX r 2X^2_{su}\colon =\int_s^u\int_s^{r_2} dX_{r_1}\otimes dX_{r_2},…, X su := s u s r ... s r 2dX r 1...dX r X^\ell_{su}\colon =\int_s^u \int_s^{r_\ell}...\int_s^{r_2} dX_{r_1}\otimes...\otimes dX_{r_\ell} where =p\ell=\lfloor p \rfloor for s,u[0,t]s,u\in[0,t] then the following “corrected Riemann sum” converges almost surely as is called the rough integral:

lim n k=0 nf(X t k)X t kt k+1 1+Df(X t k)X t kt k+1 2+...+D 1f(X t k)X t kt k+1 := 0 tf(X S)dX s\lim_{n\to\infty}\sum_{k=0}^{n} f(X_{t_k} )X^1_{t_k t_{k+1}}+Df(X_{t_k})X^2_{t_k t_{k+1}}+...+D^{\ell-1}f(X_{t_k})X^{\ell}_{t_k t_{k+1}}\colon = \int_0^t f(X_S)\otimes d\mathbf{X}_s

The collection X=(X 1,...,X )\mathbf{X}=(X^1,...,X^\ell) is called a rough path of order \ell. We say that (f(X),Df(X),...,D 1f(X))(f(X),Df(X),...,D^{\ell-1}f(X)) is controlled by X\mathbf{X}.

Example

Fractional Brownian motion (fBm), B tB_t, is a centered, continuous Gaussian process with covariance

E(B tB s)=12(t 2H+s 2H|ts| 2H)E(B_t B_s)=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right)

where H(0,1)H\in(0,1) is called the Hurst parameter. When H=1/2H=1/2 we recover regular Brownian motion.

We have that B tC pB_t\in C^p iff p>1/Hp\gt 1/H and that fBm is not a semimartingale unless H=1/2H=1/2. So SDEs driven by fBm with H<1/2H\lt 1/2 are not solvable by classical means and rough paths must be used.

References

Last revised on March 26, 2024 at 20:19:43. See the history of this page for a list of all contributions to it.